Question

What discount rate would make you indifferent between receiving $3,282.00 per year forever and $5,840.00 per year for 24.00 years? Assume the first payment of both cash flow streams occurs in one year.

Answer 1

To find the discount rate that makes you indifferent between the two cash flow streams, we need to set the present value (PV) of both cash flows equal.

1. **Cash Flow Stream 1**: An annuity of $3,282 per year forever. The present value of a perpetuity can be calculated using the formula:

\[
PV = \frac{C}{r}
\]

where \(C\) is the cash flow per period, and \(r\) is the discount rate.

So, for the perpetuity:

\[
PV_1 = \frac{3,282}{r}
\]

2. **Cash Flow Stream 2**: An annuity of $5,840 per year for 24 years. The present value of a finite annuity can be computed using the formula:

\[
PV = C \times \left(1 - (1 + r)^{-n}\right) \div r
\]

where \(C\) is the cash flow per period, \(r\) is the discount rate, and \(n\) is the number of periods.

So, for the annuity:

\[
PV_2 = 5,840 \times \left(1 - (1 + r)^{-24}\right) \div r
\]

3. **Setting the Present Values Equal**:
We need to set these two present values equal to find \(r\):

\[
\frac{3,282}{r} = 5,840 \times \left(1 - (1 + r)^{-24}\right) \div r
\]

Since \(r\) is in the denominator on both sides, we can multiply through by \(r\) (assuming \(r \neq 0\)) to simplify:

\[
3,282 = 5,840 \times \left(1 - (1 + r)^{-24}\right)
\]

4. **Solving for r**:
Now, we can simplify this equation:

\[
3,282 = 5,840 - 5,840 \times (1 + r)^{-24}
\]

Rearranging gives us:

\[
5,840 \times (1 + r)^{-24} = 5,840 - 3,282
\]

\[
5,840 \times (1 + r)^{-24} = 2,558
\]

We divide both sides by 5,840:

\[
(1 + r)^{-24} = \frac{2,558}{5,840}
\]

Calculating the right-hand side:

\[
\frac{2,558}{5,840} \approx 0.437
\]

Now, we flip both sides:

\[
1 + r = \left(0.437\right)^{-1/24}
\]

Taking the inverse gives:

\[
1 + r \approx \left(2.287\right)^{1/24}
\]

Calculating this using a calculator or estimation methods:

\[
1 + r \approx 1.1009
\]

Thus,

\[
r \approx 1.1009 - 1 \approx 0.1009 \text{ or } 10.09\%
\]

So, the discount rate that would make you indifferent between receiving $3,282 per year forever and $5,840 per year for 24 years is approximately **10.09%**.